David Cameron has been given a stark warning from Washington that dissident terrorists will be "emboldened" to intensify their attacks in Northern Ireland and millions of dollars of US investment will be threatened if he fails to persuade the Tories' Ulster Unionist partners to endorse the final stage of the peace process.

In a sign of deep unease in Washington at the Tories' electoral pact with the UUP, a bipartisan group of US Congress members have demanded Cameron must "aggressively" encourage his partners to endorse the deal in a vote in the Northern Ireland assembly on Tuesday.

"At this defining moment in the peace process, it is crucial that all the political parties speak with one voice about their shared future," the members of Congress wrote in a private letter to Cameron.

Without a unanimous vote, they said, the province's elected officials would be sending the wrong message to the people they represent, and to the world.

"Dissidents are trying to destabilise the political institutions and turn the clock back … These dissidents will continue to be emboldened if they sense there is no political unanimity on the way forward … The challenge of bringing good jobs to Northern Ireland will be made more difficult if potential investors do not believe there is political stability."

The letter by congressional leaders of the 41 million-strong Irish-American community has been passed to the Guardian before the crucial vote on last month's Hillsborough agreement between Sinn Féin and the Democratic Unionists.

A yes vote would see policing and criminal justice powers devolved to Belfast next month in what the British government has dubbed the final piece in the jigsaw of the 1998 Good Friday agreement.

The congressmen and women, whose views reflect the private misgivings of the US secretary of state, Hillary Clinton, wrote to Cameron after learning that his electoral partners in Northern Ireland were voicing doubts about endorsing the deal. Clinton is said to be alarmed that Cameron has either failed, or not bothered, to warn the UUP of the dangers of obstructing the deal.

A Tory source said: "David Cameron has consistently made clear that we support the devolution of policing and criminal justice powers … We welcomed the agreement between the DUP and Sinn Féin. The final details are for the parties in the executive, working as a four-party coalition, to decide. We hope that between now and Tuesday any outstanding issues can be solved."

The Hillsborough agreement was brokered last month under a system endorsed by the UUP when it was the largest party in Northern Ireland. This says that any change must have the consent of 50% of both the nationalist and Unionist communities and 50% of the assembly.

Sir Reg Empey, the UUP leader who was one of the architects of the Good Friday agreement, will meet his party's executive tomorrow night to decide how to vote.

The deal could still be passed tomorrow without the support of the UUP because the DUP's 36 assembly members account for more than 50% of the unionist bloc. But there are fears in Washington that a no vote from the UUP, seen as a moribund force until Cameron breathed new life into it, would destabilise many members of the DUP and undermine support for the devolution of policing and criminal justice powers.